Tian's properness conjectures and Finsler geometry of the space of Kahler metrics

By Tam'as Darvas, Yanir A. Rubinstein. 2015

175 citations recorded by the source corpus.

Well-known conjectures of Tian predict that existence of canonical Kahler metrics should be equivalent to various notions of properness of Mabuchi's K-energy functional. In some instances this has been verified, especially under restrictive assumptions on the automorphism group. We provide counterexamples to the original conjecture in the presence of continuous automorphisms. The construction hinges upon an alternative approach to properness that uses in an essential way the metric completion with respect to a Finsler metric and its quotients with respect to group actions. This approach also allows us to formulate and prove new optimal versions of Tian's conjecture in the setting of smooth and singular Kahler-Einstein metrics, with or without automorphisms, as well as for Kahler-Ricci solitons. Moreover, we reduce both Tian's original conjecture (in the absence of automorphisms) and our modification of it (in the presence of automorphisms) in the general case of constant scalar curvature metrics to a conjecture on regularity of minimizers of the K-energy in the Finsler metric completion. Finally, our results also resolve Tian's conjecture on the Moser-Trudinger inequality for Fano manifolds with Kahler-Einstein metrics.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/9dc6cdd74177604c797108a0e1a16e9cc7e11b7e

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Citations in the corpus

Monotonicity of nonpluripolar products and complex Monge–Ampère equations with prescribed singularity (2017, methodology): “4) does not always have a solution. More importantly, solvability of this equation is tied together with existence of KE metrics on Fano manifolds. It would be interesting to see if the techniques of [DR17] apply to give characterizations for existence of KE metrics with prescribed singularity type ”
Geodesic rays and stability in the cscK problem (2020, methodology): “Applying Darvas-Rubinstein’s principle [35] as in the way as [49, Theorem 3.”
C1,1 regularity for degenerate complex Monge–Ampère equations and geodesic rays (2017, background): “For background material and further references on geodesic rays, their regularity and relation to test configuration and K-stability and we refer the reader to [3, 4, 5, 11, 12, 15, 20, 21, 23, 40, 32, 33, 34, 35, 36, 37, 38, 39, 41].”
On the constant scalar curvature Kähler metrics, general automorphism group (2018, methodology): “The converse direction has been established by [5] and [25].”
On the Yau‐Tian‐Donaldson Conjecture for Generalized Kähler‐Ricci Soliton Equations (2020, methodology): “Based on the works in [22, 27], we know that the solvability of this equation is equivalent to appropriate coercivity condition of associated energy functionals.”
A variational approach to the Yau–Tian–Donaldson conjecture (2015, methodology): “sion of the classical Dirichlet functional. Note that (iii)=)(i) can alternatively be established using a variational argument [BBEGZ11], and that (i)=)(ii) was recently given a very elegant proof in [DR15], based on new ideas that have had a strong in uence on the strategy of proof of Theorem A pro”
G-uniform stability and Kähler–Einstein metrics on Fano varieties (2019, methodology): “18 follows from the works of Darvas and Hisamoto via the general framework by Darvas-Rubinstein (in [33]) for proving Tian’s properness conjecture from [76].”
Geometric pluripotential theory on Kähler manifolds (2019, methodology): “Rubinstein to verify numerous related conjectures of Tian [56] going back to the nineties [105, 106].”
On the constant scalar curvature Kähler metrics (II)—Existence results (2020, methodology): “Rubinstein conjectured in [33](Conjecture 2.”